Math 2.0

Associative rings and algebras Definition

Quotient ring

Builds on Two-sided idealRing

Definition

Let RR be a ring and let II be a two-sided ideal of RR. For a∈Ra \in R, the coset of aa is a+I={a+x:x∈I}a + I = \{a + x : x \in I\}. The quotient ring R/IR/I is the set of the cosets of II, with these operations: (a+I)+(b+I)=(a+b)+I,(a+I)(b+I)=ab+I.(a + I) + (b + I) = (a + b) + I, \qquad (a + I)(b + I) = a b + I. These operations do not depend on the choice of aa and bb in their cosets. With them, R/IR/I is a ring with zero 0+I=I0 + I = I and identity 1+I1 + I.

Remarks

Two elements a,b∈Ra, b \in R have the same coset if and only if a−b∈Ia - b \in I. The map π ⁣:R→R/I\pi \colon R \to R/I, π(a)=a+I\pi(a) = a + I, is a surjective ring homomorphism with kernel II. For n≥1n \ge 1, the quotient ℤ/nℤ\mathbb{Z}/n\mathbb{Z} is the ring of integers modulo nn. The quotient R/RR/R is the zero ring.

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Group, concept, no evidence levelGroupconceptRing, definition, no evidence levelRingdefinitionIdeal, definition, no evidence levelIdealdefinitionTwo-sided ideal, definition, no evidence levelTwo-sided idealdefinitionQuotient ring, definition, no evidence levelQuotient ringdefinition
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  1. Group concept
  2. Ring definition
  3. Ideal definition
  4. Two-sided ideal definition
  5. Quotient ring definition

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Further reading

  1. Web https://en.wikipedia.org/wiki/Quotient_ring, Formal quotient ring construction
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DefinesQuotient ring
UsesTwo-sided ideal, Ring

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